Transcript Sequence analysis course

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Introduction to bioinformatics
2008
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Lecture 12
Phylogenetic methods
Tree distances
Evolutionary (sequence distance) = sequence dissimilarity
human
5
x
human
1
mouse
6
x
fugu
7
3
x
Drosophila
14
10
9
mouse
2
1
1
x
fugu
6
Drosophila
Note that with evolutionary methods for generating trees you get distances
between objects by walking from one to the other.
Phylogeny methods
1. Distance based – pairwise distances (input is
distance matrix)
2. Parsimony – fewest number of evolutionary events
(mutations) – relatively often fails to reconstruct
correct phylogeny, but methods have improved
recently
3. Maximum likelihood – L = Pr[Data|Tree] – most
flexible class of methods - user-specified
evolutionary methods can be used
Similarity criterion for phylogeny
• A number of methods (e.g. ClustalW) use sequence
identity with Kimura (1983) correction:
Corrected K = - ln(1.0-K-K2/5.0), where K is percentage
divergence corresponding to two aligned sequences
• There are various models to correct for the fact that
the true rate of evolution cannot be observed through
nucleotide (or amino acid) exchange patterns (e.g.
back mutations)
• Saturation level is ~94% changed sequences, higher
real mutations are no longer observable
Distance based --UPGMA
Let Ci and Cj be two disjoint clusters:
1
di,j = ———————— pq dp,q, where p  Ci and q  Cj
|Ci| × |Cj|
Ci
Cj
In words: calculate the average over all pairwise
inter-cluster distances
Clustering algorithm: UPGMA
Initialisation:
•
Fill distance matrix with pairwise distances
•
Start with N clusters of 1 element each
Iteration:
1. Merge cluster Ci and Cj for which dij is minimal
2. Place internal node connecting Ci and Cj at height dij/2
3. Delete Ci and Cj (keep internal node)
Termination:
•
When two clusters i, j remain, place root of tree at height dij/2
d
Ultrametric Distances
•A tree T in a metric space (M,d) where d is ultrametric
has the following property: there is a way to place a root
on T so that for all nodes in M, their distance to the root
is the same. Such T is referred to as a uniform
molecular clock tree.
•(M,d) is ultrametric if for every set of three elements
i,j,k∈M, two of the distances coincide and are greater
than or equal to the third one (see next slide).
•UPGMA is guaranteed to build correct
tree if distances are ultrametric. But it fails
if not!
Ultrametric Distances
Given three leaves, two distances are equal
while a third is smaller:
d(i,j)  d(i,k) = d(j,k)
a+a  a+b = a+b
i
a
b
a
j
k
nodes i and j are at same
evolutionary distance from k
– dendrogram will therefore
have ‘aligned’ leafs; i.e. they
are all at same distance
from root
No need to memorise formula
Evolutionary clock speeds
Uniform clock: Ultrametric
distances lead to identical
distances from root to leafs
Non-uniform evolutionary clock: leaves have different
distances to the root -- an important property is that of
additive trees. These are trees where the distance between
any pair of leaves is the sum of the lengths of edges
connecting them. Such trees obey the so-called 4-point
condition (next slide).
Additive trees
All distances satisfy 4-point condition:
For all leaves i,j,k,l:
d(i,j) + d(k,l)  d(i,k)
+ d(j,l)
=
d(i,l) + d(j,k)
(a+b)+(c+d)  (a+m+c)+(b+m+d) = (a+m+d)+(b+m+c)
k
i
a
c
m
j
b
d
l
Result: all pairwise distances obtained by traversing
No need to memorise formula
the tree
Additive trees
In additive trees, the distance between any pair
of leaves is the sum of lengths of edges
connecting them
Given a set of additive distances: a unique tree T
can be constructed:
•For two neighbouring leaves i,j with common
parent k, place parent node k at a distance
from any node m with
d(k,m) = ½ (d(i,m) + d(j,m) – d(i,j))
i
c
= ½ ((a+c) + (b+c) – (a+b))
No need to memorise formula
a
b
j
c
k
m
Utrametric/Additive distances
If d is ultrametric then d is additive
If d is additive it does not follow that d is
ultrametric
Can you prove the first statement?
Distance based -Neighbour
joining (Saitou and Nei, 1987)
• Widely used method to cluster DNA
or protein sequences
• Global measure – keeps total branch
length minimal, tends to produce a
tree with minimal total branch length
(concept of minimal evolution)
• Agglomerative algorithm
• Leads to unrooted tree
Neighbour-Joining (Cont.)
• Guaranteed to produce correct tree if
distances are additive
• May even produce good tree if
distances are not additive
• At each step, join two nodes such
that total tree distances are minimal
(whereby the number of nodes is
decreased by 1)
Neighbour-Joining
• Contrary to UPGMA, NJ does not assume taxa to be
equidistant from the root
• NJ corrects for unequal evolutionary rates between
sequences by using a conversion step
• This conversion step requires the calculation of
converted (corrected) distances, r-values (ri) and
transformed r values (r’i), where ri = dij and r’i = ri /(n2), with n each time the number of (remaining) nodes in
the tree
• Procedure:
– NJ begins with an unresolved star tree by joining all taxa onto
a single node
– Progressively, the tree is decomposed (star decomposition),
by selecting each time the taxa with the shortest corrected
distance, until all internal nodes are resolved
Neighbour joining
x
y
y
y
x
(a)
x y
(d)
(c)
(b)
z
y
x
(e)
(f)
At each step all possible ‘neighbour joinings’ are checked and the one
corresponding to the minimal total tree length (calculated by adding all
branch lengths) is taken.
Neighbour joining – ‘correcting’
distances
Finding neighbouring leaves:
Define
d’ij = dij – ½ (ri + rj)
[d’ij is corrected distance]
Where
ri = k dik and
1
r’i =
——— k dik
|L| - 2
[ |L| is current number of nodes]
Total tree length Dij is minimal iff i and j are
neighbours
No need to memorise
Algorithm: Neighbour joining
Initialisation:
•Define T to be set of leaf nodes, one per sequence
•Let L = T
Iteration:
•Pick i,j (neighbours) such that d’i,j is minimal (minimal total tree
length) [this does not mean that the OTU-pair with smallest
uncorrected distance is selected!]
•Define new ancestral node k, and set dkm = ½ (dim + djm – dij) for
all m  L
•Add k to T, with edges of length dik = ½ (dij + r’i – r’j)
•Remove i,j from L; Add k to L
Termination:
•When L consists of two nodes i,j and the edge between them of
length dij
No need to memorise, but know how NJ works intuitively
Algorithm: Neighbour joining
NJ algorithm in words:
1. Make star tree with ‘fake’ distances (we need these to be
able to calculate total branch length)
2. Check all n(n-1)/2 possible pairs and join the pair that leads
to smallest total branch length. You do this for each pair by
calculating the real branch lengths from the pair to the
common ancestor node (which is created here – ‘y’ in the
preceding slide) and from the latter node to the tree
3. Select the pair that leads to the smallest total branch length
(by adding up real and ‘fake’ distances). Record and then
delete the pair and their two branches to the ancestral node,
but keep the new ancestral node. The tree is now 1 one node
smaller than before.
4. Go to 2, unless you are done and have a complete tree with
all real branch lengths (recorded in preceding step)
Parsimony & Distance
Sequences
Drosophila
fugu
mouse
human
1
t
a
a
a
2
t
a
a
a
3
a
t
a
a
4
t
t
a
a
5
t
t
a
a
6
a
a
t
a
human
x
mouse
2
x
fugu
4
4
x
Drosophila
5
5
3
7
a
a
a
t
parsimony
Drosophila
1
4
2
fugu
Drosophila
5
3
mouse
6
7
human
distance
mouse
2
1
2
1
x
fugu
1
human
Problem: Long Branch Attraction
(LBA)
• Particular problem associated with parsimony
methods
• Rapidly evolving taxa are placed together in a tree
regardless of their true position
• Partly due to assumption in parsimony that all
lineages evolve at the same rate
• This means that also UPGMA suffers from LBA
• Some evidence exists that also implicates NJ
A
A
B
C
True tree
D
B
C
D
Inferred tree
Maximum likelihood
Pioneered by Joe Felsenstein
• If data=alignment, hypothesis = tree, and under a given
evolutionary model,
maximum likelihood selects the hypothesis (tree) that
maximises the observed data
• A statistical (Bayesian) way of looking at this is that the tree
with the largest posterior probability is calculated based on the
prior probabilities; i.e. the evolutionary model (or
observations).
• Extremely time consuming method
• We also can test the relative fit to the tree of different models
(Huelsenbeck & Rannala, 1997)
Maximum likelihood
Methods to calculate ML tree:
• Phylip (http://evolution.genetics.washington.edu/phylip.html)
• Paup (http://paup.csit.fsu.edu/index.html)
• MrBayes (http://mrbayes.csit.fsu.edu/index.php)
Method to analyse phylogenetic tree with ML:
• PAML (http://abacus.gene.ucl.ac.uk/software/paml.htm)
The strength of PAML is its collection of sophisticated substitution models to
analyse trees.
• Programs such as PAML can test the relative fit to the
tree of different models (Huelsenbeck & Rannala, 1997)
Maximum likelihood
• A number of ML tree packages (e.g. Phylip, PAML)
contain tree algorithms that include the assumption of a
uniform molecular clock as well as algorithms that don’t
• These can both be run on a given tree, after which the
results can be used to estimate the probability of a
uniform clock.
How to assess confidence in tree
How to assess confidence in tree
• Distance method – bootstrap:
– Select multiple alignment columns with
replacement (scramble the MSA)
– Recalculate tree
– Compare branches with original (target) tree
– Repeat 100-1000 times, so calculate 1001000 different trees
– How often is branching (point between 3
nodes) preserved for each internal node in
these 100-1000 trees?
– Bootstrapping uses resampling of the data
The Bootstrap -- example
Original
1
M
M
2
C
A
C
3
V
V
L
4
K
R
R
5
V
L
2x
3
V
Scrambled V
L
4
K
R
R
3
V
V
L
6
I
I
L
7
Y
F
F
8
S
S
T
8
S
S
T
6
I
I
L
Used multiple times in
resampled (scrambled)
MSA below
5
1
2
3
4
3x
8
S
S
T
6
I
I
L
6
I
I
L
Only boxed alignment columns are randomly selected in this example
1
1
2
5
3
Nonsupportive
Some versatile phylogeny software
packages
• MrBayes
• Paup
• Phylip
MrBayes: Bayesian Inference of
Phylogeny
• MrBayes is a program for the Bayesian estimation of phylogeny.
• Bayesian inference of phylogeny is based upon a quantity called the
posterior probability distribution of trees, which is the probability of
a tree conditioned on the observations.
• The conditioning is accomplished using Bayes's theorem. The
posterior probability distribution of trees is impossible to calculate
analytically; instead, MrBayes uses a simulation technique called
Markov chain Monte Carlo (or MCMC) to approximate the posterior
probabilities of trees.
• The program takes as input a character matrix in a NEXUS file
format. The output is several files with the parameters that were
sampled by the MCMC algorithm. MrBayes can summarize the
information in these files for the user.
No need to memorise
MrBayes: Bayesian Inference of
Phylogeny
MrBayes program features include:
• A common command-line interface for Macintosh, Windows, and UNIX
operating systems;
• Extensive help available via the command line;
• Ability to analyze nucleotide, amino acid, restriction site, and morphological
data;
• Mixing of data types, such as molecular and morphological characters, in a
single analysis;
• A general method for assigning parameters across data partitions;
• An abundance of evolutionary models, including 4 X 4, doublet, and codon
models for nucleotide data and many of the standard rate matrices for amino
acid data;
• Estimation of positively selected sites in a fully hierarchical Bayes framework;
• The ability to spread jobs over a cluster of computers using MPI (for Macintosh
and UNIX environments only).
No need to memorise
PAUP
Phylip – by Joe Felsenstein
Phylip programs by type of data
• DNA sequences
• Protein sequences
• Restriction sites
• Distance matrices
• Gene frequencies
• Quantitative characters
• Discrete characters
• tree plotting, consensus trees, tree distances and tree
manipulation
http://evolution.genetics.washington.edu/phylip.html
Phylip – by Joe Felsenstein
Phylip programs by type of algorithm
• Heuristic tree search
• Branch-and-bound tree search
• Interactive tree manipulation
• Plotting trees, consenus trees, tree distances
• Converting data, making distances or bootstrap
replicates
http://evolution.genetics.washington.edu/phylip.html
The Newick tree format
C
A
Ancestor1
5
3
4
E
D
B
6
5
11
(B,(A,C,E),D); -- tree topology
root
(B:6.0,(A:5.0,C:3.0,E:4.0):5.0,D:11.0); -- with branch lengths
(B:6.0,(A:5.0,C:3.0,E:4.0)Ancestor1:5.0,D:11.0)Root;
-- with branch lengths and ancestral node names
Distance methods: fastest
• Clustering criterion using a distance matrix
• Distance matrix filled with alignment scores
(sequence identity, alignment scores, Evalues, etc.)
• Cluster criterion
Kimura’s correction for protein
sequences (1983)
This method is used for proteins only. Gaps are ignored and only
exact matches and mismatches contribute to the match score.
Distances get ‘stretched’ to correct for back mutations
S = m/npos,
Where m is the number of exact matches and
npos the number of positions scored
D = 1-S
Corrected distance = -ln(1 - D - 0.2D2)
(see also
earlier slide)
Reference:
M. Kimura, The Neutral Theory of Molecular Evolution, Camb.
Uni. Press, Camb., 1983.
Sequence similarity criteria for
phylogeny
• In
addition to the Kimura correction, there are
various models to correct for the fact that the true
rate of evolution cannot be observed through
nucleotide (or amino acid) exchange patterns (e.g.
due to back mutations).
• Saturation level is ~94%, higher real mutations
are no longer observable
A widely used protocol to infer
a phylogenetic tree
• Make an MSA
• Take only gapless positions and
calculate pairwise sequence distances
using Kimura correction
• Fill distance matrix with corrected
distances
• Calculate a phylogenetic tree using
Neigbour Joining (NJ)
Phylogeny disclaimer
• With all of the phylogenetic methods,
you calculate one tree out of very many
alternatives.
• Only one tree can be correct and depict
evolution accurately.
• Incorrect trees will often lead to ‘more
interesting’ phylogenies, e.g. the whale
originated from the fruit fly etc.
Take home messages
• Rooted/unrooted trees, how to root a tree
• Make sure you can do the UPGMA algorithm
and understand the basic steps of the NJ
algorithm
• Understand the three basic classes of
phylogenetic methods: distance-based,
parsimony and maximum likelihood
• Make sure you understand bootstrapping (to
asses confidence in tree splits)